if A is infinite and B is a finite subset of A⁢\tmspace+.1667⁢e⁢m⁢\tmspace-.1667⁢e⁢m, then A∖B is infinite

Proof. The proof is by contradiction. If A∖B
would be finite, there would exist a k∈ℕ and a bijectionf:{1,…,k}→A∖B. Since B is finite, there
also exists a bijection g:{1,…,l}→B. We can then define
a mappingh:{1,…,k+l}→A by

h⁢(i)

=

{f⁢(i)when⁢i∈{1,…,k},g⁢(i-k)when⁢i∈{k+1,…,k+l}.

Since f and g are bijections, h is a bijection between
a finite subset of ℕ and A. This is a contradiction
since A is infinite. □

Title

if A is infinite and B is a finite subset of A⁢\tmspace+.1667⁢e⁢m⁢\tmspace-.1667⁢e⁢m, then A∖B is infinite